NMR relaxation time distribution measurements in porous media are commonly undertaken to determine the pore size distribution and play an important role in the characterisation of porous media including permeability, wettability, capillary pressure, residual oil saturation and gas volume [1-3]. The T2 relaxation time distribution measurement is the essential basis of most downhole NMR logging measurements [4-10]. The T2 distribution measurement in this case is a bulk measurement from a region of space defined by the magnet and RF probe geometry.
Relaxation time distribution studies in the literature are almost entirely bulk measurements despite the fact that reservoir rocks, and reservoir core plugs, are frequently macroscopically heterogeneous. T2 relaxation time distribution mapping is highly desirable because the bedding plane structure, ubiquitous in sedimentary rocks, and frequently finer scale than the core plug itself, will often result in different pore properties within the sample.
Commonly employed multi-echo T2 mapping sequences with frequency encode gradients are not suitable for this purpose because of the inherently short relaxation times [11, 12] and strong susceptibility contrast [13] in reservoir rocks. Pure phase encode techniques are robust in their ability to generate true fluid content images and relaxation mappings in porous media [14-17]. Li measured 1D spatially resolved T2 distributions with separate phase encoding of each echo in a multi-echo CPMG pulse train [15]. This method is not optimal in terms of gradient duty cycle and gradient stabilization, as phase encoding and phase unwinding gradients are required for each echo. Petrov et al [16] improved the Spin Echo Single Point Imaging (SE SPI) method by restricting phase encoding to the first pulse interval preceding readout through multiple refocusing. A CPMG prepared SPRITE (Single Point Ramped Imaging with T1 Enhancement) sequence for T2 distribution mapping has also been proposed [17].
The SE SPI experiment has great utility for routine rock core plug measurements, but has an acquisition time that is proportional to the number of k-space points. This makes simple Cartesian sampling unrealistic for the 2D case. Petrov et al employed Compressing Sensing (CS) [17] to mitigate the problem. In the current work, optimal k-space sampling schemes are developed to improve the measurement time.
Undersampling k-space has been a popular topic in MRI research for decades [18-20]. Compressed sensing reconstruction has been successfully applied to MRI [21] and may achieve high acceleration factors. However, the calculations can be time consuming and CS sometimes yields unreliable results, especially for low contrast features.
Geometric k-space sampling patterns are very natural and are routinely employed in centric scan SPRITE [22, 23]. Sampling patterns which utilize radial, spiral, conical or sectoral trajectories omit the extremes of k-space in a Cartesian representation. These omitted points are assigned a value of zero. This approach is simple and reliable but saves only 20% of the k-space data points in 2D, resulting in minor reductions in acquisition time.
Spin-echo single point imaging has been employed for one dimensional (“1D”) T2 distribution mapping, but a simple extension to two-dimensional (“2D”) is challenging since the time increase is n fold, where n is the number of pixels in the second dimension. Nevertheless 2D T2 mapping in fluid saturated rock core plugs is highly desirable because the bedding plane structure in rocks often results in different pore properties within the sample.